Fermat

1
Fermats Last Theorem can Decode Nazi military
Ciphers

• What does Euclid, Pythagoras, Pierre de Fermat,
  Sophie Germain, and Lame all have in common?

2
The connection between Fermats Last Theorem and
Nazi military ciphers are

• Cracking the World War II, circa mid 1940s,
  ciphers used derivatives of mathematics
  developed by number theory connected to Fermats
  Last Theorem, which was x3y3z3 developed in
  the 17th century.
• Fermats theorem, however, borrowed the best
  equation from the 6th century B.C. to measure a
  triangle in the history of mathematics which was
  Pythagoras geometric theorem a2b2c2 as
  his premise.
• After Fermat developed his equation he began to
  substitute the exponents from 3 to 4,5,6 on up,
  and continued to use the mathematical method
  called trial and error.

3

• Fermats proof had to prove that these
  substitutions had no solutions that existed
  within this infinity of infinities.
• And even though the time period between these 2
  events are 302 years apart, this type of logic
  parallels with the WWII Bletchley Park military
  headquarters in the UK when they were trying to
  crack the secret war codes using some form of
  deductive reasoning which stems from Euclids
  geometric laws.

4
What happens next historically in Mathematics?

• Fermat declared that he had written his proof
  down for this number theory, which was never
  found or proven.
• So a new mathematician enters the male dominated
  arena, in the early 19th century, by the name of
  Sophie Germain. Sophie felt compelled that it
  was her obligation to rediscover his proof.
• She started working with prime numbers whose
  numbers have no divisors, and began to develop
  pairs of numbers like factorisation, I.e.
  111x11 or (2x5)111.
• Miss Germain subtituted n as an exponent into
  Fermats equation which became xnynzn.

5

• Due to tight restrictions of these prime numbers
  for n forced her to prove there could be no
  solutions which backed Fermats theory.
• This is as far as her contributions went because
  without Professor Carl Friedrich Guass, women
  didnt have any academic place in the mathematics
  world.
• See she was forced to take on the identity of a
  man by using the name Monsieur LeBlanc to be able
  to submit her mathematical number theories at
  university level. Only then when Professor Guass
  discovered him as a her did the submission of her
  mathematical number theories enable her to grow
  as a mathematician.

6

• At this juncture in time Professor Guass switches
  to the astronomy department, and the lack of
  written communication weakened the never married
  Sophies confidence to continue her studies in
  pure mathematics.
• Since Guass was a great stepping stone and the
  fact that the stating of values of n remained
  intractable made her later concentrate on the
  modern theory of elasticity.
• Before Miss Germain died of breast cancer she
  received an honorary award degree from Professor
  Guass through Gottingen for her extensive
  research. This is the same university where
  Sonia Kovalevsky in 1874 was the first woman to
  receive a Phd in mathematics.

7
Lets Examine Military Ciphers

• The WWII Bletchley Park files concluded that the
  British military chose people who had a good
  sense of crosswords puzzles using a keen sense of
  numerical patterns.
• However, it was extremely difficult to examine
  lines of letters that absolutely no sense.
• The key ingredient here to decoding was the
  typewriter machine called the Enigma which
  translated and printed these random sequence
  codes using a 3 wheel machine with 4 rotors.

8

• This machine had double indicators enciphering to
  get this Morse code (radio transmissions) sent
  abroad via German military.
• Therefore, the only man to break such a code in
  1944 was a German-Jewish mathematician by the
  name Alan Turing, from Cambridge.
• Turing used deducted reasoning substituting pairs
  of numbers , 3 letters at a time, forming these
  secret tables.

9

• In conjunction with the Lorenz machine(stolen
  from the German navy by the British), the modulo
  2 addition mathematical system which transmits a
  string of letters which then becomes mixed up
  during transmission, across seas, then prints the
  code correctly after transmission.
• This process was the breaking point for Turings
  research.
• This code is now referred to as algorithms in
  mathematics.

10

• Algorithms as such are used in either 1038 or
  40-128 bit encryptions for safe online shopping
  or banking.
• Bletchley Park would be considered the global
  hackers of the 21st century sniffing for online
  passwords and breaking into peoples email
  accounts.
• This type of coding could of even been helpful to
  Mary Queen of Scots, who tried military espionage
  on Queen Elizabeth. Mary eventually became
  trapped by her own Beale cipher codes.

11

• Such strategy later influenced numerical
  strategies in WWI and WWII.
• Hence, Elizabeth had Mary killed because the
  hidden location of the gold fortune buried in
  Virginia, circa 19th century, was never found and
  remains a mystery to date.

12
Decoding of the Nazi Secrets.

• Our practice cipher is Bo fbtz djqifs up csfbl!
  This phrase was a simple switching positions of
  the alphabet letters, which was (n-1), therefore,
  it would be the previous alphabet letter. This
  above phrase translates to An easy cipher to
  break! Such success of a translation entices
  one to prepare for more challenging ciphers.
• Now lets tackle a 5x5 letter group divided into
  7 columns, instead of one line of string. The
  example is GEGOH. It identifies to a 5x5
  matrices from pre-calculus, however what does a
  mathematician do with 2 digit numbers
  (G6,E5,G6,O16,H8) and how could one
  possibly finish a 5x5 matrix within 2 steps?

13

• Other examples of matrices used were 3x3s within
  a four to five sentence.
• Basically cracking the ciphers uses applied
  random movement of how row swapping solves
  algorithms producing congruent answers.
  Geometric deductions cribbed the dragging of
  digraphs, hence, the pattern repeated itself 2
  times with 3 pairs between them.

14
What to understand.

• Understanding such logic and number theory
  connects to Euclid, Pythagoras, Fermat, Sophie
  Germain, and Lame.
• In number theory, from the equation a2b2c2
  to xnynzn, the prime number 3(Euclid proved
  cubes), 5(Sophie proved, as well as 2n/-1),
  7(Lame proved) all worked except for the exponent
  number 4(1x4,2x2) because it was not a prime
  number.

15

• It was not a coincidence that the numbers 3,5,7
  were all prime numbers or that they matched the
  ciphers of 3x3s, 5x5s in 7 columns or the
  double equation of the alphanumeric number minus
  1.
• All of this finally fits the rules of
  factorisation.
• Gabriel Lame was an applied mathematician and was
  led to Fermats Last Theorem like Sophie Germain.
  
• Lame made a substantial contribution to the
  problem ab is xnynan by solving the case
  n7.
• Although he believed he had solved the whole
  problem in one stage he later overlooked the lack
  of unique factorisation.

16

• Lame did important contributions working on
  differential geometry, number theory and showed
  that the number of the divisions in Euclidean
  algorithms never exceeded 5x the number of digits
  in the smaller number.
• In 1839, Lamed proved Fermats theorem that n7.
• In 1753, Euclid proved p23q2 even though his
  numbers didnt behave, his case study n3, did
  work.
• Sophie proved her case study that if n an 2n1
  are primes then xnynzn implies one of the
  x,y,z is divisible by n with numbers lt100 and all
  numbers lt197.

17

• Her case study split Fermats theorem case 1
  that none of x, y, z is divisible by n. And
  case2 that 1and only(prime number )1 of x,y,z is
  divisible by n.
• Sophies proof that the number 5 splits itself
  into 2(ie 221 or 22 or 1x5) shows that an even
  number plus one divisible number by 5 are
  distinct.
• Euclids cube using the number 3 as his case
  study proved that pa3-9ab2, q3(a2b-b3) then
  p23q2(a23b2)3 is a cube an ab exist as
  pqa(b times the square root of 3(complex
  factorisation number of i). Therefore, that is
  how Fermats Last Theorem bridges to decoding
  Nazi secrets.